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   "cell_type": "markdown",
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   "source": [
    "# 线性回归的从零开始实现\n",
    "\n",
    "在了解了线性回归的背景知识之后，现在我们可以动手实现它了。尽管强大的深度学习框架可以减少大量重复性工作，但若过于依赖它提供的便利，会导致我们很难深入理解深度学习是如何工作的。因此，本节将介绍如何只利用`Tensor`和`GradientTape`来实现一个线性回归的训练。\n",
    "首先，导入本节中实验所需的包或模块，其中的matplotlib包可用于作图，且设置成嵌入显示。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "1"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from IPython import display\n",
    "from matplotlib import pyplot as plt\n",
    "import tensorflow as tf\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 生成数据集\n",
    "\n",
    "我们构造一个简单的人工训练数据集，它可以使我们能够直观比较学到的参数和真实的模型参数的区别。设训练数据集样本数为1000，输入个数（特征数）为2。给定随机生成的批量样本特征$\\boldsymbol{X} \\in \\mathbb{R}^{1000 \\times 2}$，我们使用线性回归模型真实权重$\\boldsymbol{w} = [2, -3.4]^\\top$和偏差$b = 4.2$，以及一个随机噪声项$\\epsilon$来生成标签\n",
    "\n",
    "$$\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{w} + b + \\epsilon,$$\n",
    "\n",
    "其中噪声项$\\epsilon$服从均值为0、标准差为0.01的正态分布。噪声代表了数据集中无意义的干扰。下面，让我们生成数据集。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "2"
    }
   },
   "outputs": [],
   "source": [
    "num_inputs = 2\n",
    "num_examples = 1000\n",
    "true_w = [2, -3.4]\n",
    "true_b = 4.2\n",
    "features = tf.random.normal(stddev=1, shape=(num_examples, num_inputs))\n",
    "labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] + true_b\n",
    "labels += tf.random.normal(stddev=0.01, shape=labels.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意，`features`的每一行是一个长度为2的向量，而`labels`的每一行是一个长度为1的向量（标量）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "3"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(<tf.Tensor: id=31, shape=(2,), dtype=float32, numpy=array([ 0.01317476, -1.5601287 ], dtype=float32)>,\n",
       " <tf.Tensor: id=35, shape=(), dtype=float32, numpy=9.521702>)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features[0], labels[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "通过生成第二个特征`features[:, 1]`和标签 `labels` 的散点图，可以更直观地观察两者间的线性关系。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "4"
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   },
   "outputs": [
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     "output_type": "display_data"
    }
   ],
   "source": [
    "def use_svg_display():\n",
    "    # 用矢量图显示\n",
    "    display.set_matplotlib_formats('svg')\n",
    "\n",
    "def set_figsize(figsize=(3.5, 2.5)):\n",
    "    use_svg_display()\n",
    "    # 设置图的尺寸\n",
    "    plt.rcParams['figure.figsize'] = figsize\n",
    "\n",
    "set_figsize()\n",
    "plt.scatter(features[:, 1].numpy(), labels.numpy(), 1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "## 读取数据\n",
    "\n",
    "在训练模型的时候，我们需要遍历数据集并不断读取小批量数据样本。这里我们定义一个函数：它每次返回`batch_size`（批量大小）个随机样本的特征和标签。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "5"
    }
   },
   "outputs": [],
   "source": [
    "# 本函数已保存在d2lzh包中方便以后使用\n",
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    random.shuffle(indices)  # 样本的读取顺序是随机的\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        j = min(i + batch_size, num_examples)\n",
    "        yield features[i:j], labels[i:j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "让我们读取第一个小批量数据样本并打印。每个批量的特征形状为(10, 2)，分别对应批量大小和输入个数；标签形状为批量大小。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "6"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tf.Tensor(\n",
      "[[ 0.01317476 -1.5601287 ]\n",
      " [-1.5447617   0.8646286 ]\n",
      " [-0.69407433  0.5979207 ]\n",
      " [-1.3370134   0.06325391]\n",
      " [-0.3141921   1.259486  ]\n",
      " [ 0.48622438  0.78185797]\n",
      " [ 0.08360149 -0.7971149 ]\n",
      " [ 0.23262888 -0.9619031 ]\n",
      " [ 0.97124684 -0.8057955 ]\n",
      " [ 0.58449775 -0.01893086]], shape=(10, 2), dtype=float32) tf.Tensor(\n",
      "[ 9.521702  -1.8393052  0.7744466  1.3168004 -0.698446   2.5151038\n",
      "  7.069128   7.93736    8.877049   5.420314 ], shape=(10,), dtype=float32)\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义模型\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "9"
    }
   },
   "outputs": [],
   "source": [
    "class LrModel:\n",
    "    def __init__(self):\n",
    "        self.w = tf.Variable(tf.random.normal(stddev=0.01, shape=(num_inputs, 1)))\n",
    "        self.b = tf.Variable(tf.zeros(shape=(1,)))\n",
    "        \n",
    "    def __call__(self, X): \n",
    "        return tf.matmul(X, self.w) + self.b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义损失函数\n",
    "\n",
    "我们使用上一节描述的平方损失来定义线性回归的损失函数。在实现中，我们需要把真实值`y`变形成预测值`y_hat`的形状。以下函数返回的结果也将和`y_hat`的形状相同。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "10"
    }
   },
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):  # 本函数已保存在d2lzh包中方便以后使用\n",
    "    return (y_hat - tf.reshape(y, y_hat.shape)) ** 2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 训练模型\n",
    "\n",
    "在训练中，我们将多次迭代模型参数。在每次迭代中，我们根据当前读取的小批量数据样本（特征`X`和标签`y`），通过调用`gradient`计算小批量随机梯度，并用优化算法`sgd`迭代模型参数。由于我们之前设批量大小`batch_size`为10，每个小批量的损失`l`的形状为(10, 1)。\n",
    "\n",
    "在一个迭代周期（epoch）中，我们将完整遍历一遍`data_iter`函数，并对训练数据集中所有样本都使用一次（假设样本数能够被批量大小整除）。这里的迭代周期个数`num_epochs`和学习率`lr`都是超参数，分别设3和0.03。在实践中，大多超参数都需要通过反复试错来不断调节。虽然迭代周期数设得越大模型可能越有效，但是训练时间可能过长。而有关学习率对模型的影响，我们会在后面“优化算法”一章中详细介绍。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "12"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.041898\n",
      "epoch 2, loss 0.000163\n",
      "epoch 3, loss 0.000051\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "model = LrModel()\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):  # 训练模型一共需要num_epochs个迭代周期\n",
    "    # 在每一个迭代周期中，会使用训练数据集中所有样本一次（假设样本数能够被批量大小整除）。X\n",
    "    # 和y分别是小批量样本的特征和标签\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        with tf.GradientTape() as t:\n",
    "            l = loss(model(X), y)  # l是有关小批量X和y的损失\n",
    "        dw, db = t.gradient(l, [model.w, model.b]) # 小批量的损失对模型参数求梯度\n",
    "        model.w.assign_sub(lr * dw / batch_size)\n",
    "        model.b.assign_sub(lr * db / batch_size)  # 使用小批量随机梯度下降迭代模型参数\n",
    "    train_l = loss(model(features), labels)\n",
    "    print('epoch %d, loss %f' % (epoch + 1, tf.reduce_mean(train_l).numpy()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "训练完成后，我们可以比较学到的参数和用来生成训练集的真实参数。它们应该很接近。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "13"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([2, -3.4], <tf.Variable 'Variable:0' shape=(2, 1) dtype=float32, numpy=\n",
       " array([[ 1.9996991],\n",
       "        [-3.3989727]], dtype=float32)>)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "true_w, model.w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "attributes": {
     "classes": [],
     "id": "",
     "n": "14"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4.2,\n",
       " <tf.Variable 'Variable:0' shape=(1,) dtype=float32, numpy=array([4.199808], dtype=float32)>)"
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     "execution_count": 11,
     "metadata": {},
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